c# – Unity, Pixel Perfect and Scaling

I am totally stuck on with pixel perfectness of the game. I have tried a few pixel perfect camera -assets, but they have lots of issues considering scalability. is it normal that these cameras zoom in and out, instead of scaling the camera screen when using different resolutions for the game. Because this is a huge problem for me, cause i need to make a game for aspect ratio of 16:10, and every resolution that has the same aspect ratio is needed to scale it correctly keeping the pixel perfectness. The ortographic camera size is dependent of screen width (width / PPU / 2) So it always zooms in or out with different resolutions.

Question is: Is there a way to make it scale the game sprites instead of zooming the view?

reference request – Jacobson-style Galois theory on perfect closure

Jacobson developed a ‘Galois’ correspondence for purely inseperable extensions of exponent 1 (only consisting of pth roots), where he showed there was a correspondence between its subfields and lie-algebras of derivations given below:
$$Fmapsto mathcal{D}_F(K)$$
$$mathcal{D’}mapsto F, text{the field of constants of} space mathcal{D’}$$
Davis extends this to exponent 2 extensions, and conjectures that higher derivation Lie-algebras should correspond to higher exponents.
Firstly, I’d like to ask for a list of books or articles that explore these ideas, especially the research done in 1968-1975 or more recently. I found the original papers in the American Mathematical Society very inaccessible, and have mostly been working of Lindsay Childs’ survey of this research, but would like to learn more.

The other question I have, is if this theory is applicable to the perfect closure of a field (in its algebraic closure, or more generally): is there a corresponding notion of an ‘absolute lie-algebra of derivations’ to that of the absolute galois group? If so please provide me with references to this too.

upper lower bounds – Is this a valid proof for $I(n^2) geq frac{5}{3}$, if $q^k n^2$ is an odd perfect number with special prime $q$?

Let $sigma(x)$ denote the sum of divisors of the positive integer $x$, and let $I(x)=sigma(x)/x$ be the abundancy index of $x$.

Note that both $sigma$ and $I$ are multiplicative functions.

A number $m$ is said to be perfect if $sigma(m)=2m$. Equivalently, $I(m)=2$.

Euler proved that an odd perfect number, if one exists, must have the form
$$m = q^k n^2$$
where $q$ is the special prime satisfying $q equiv k equiv 1 pmod 4$ and $gcd(q,n)=1$.

Since $q$ is prime, we have
$$frac{q+1}{q} = I(q) leq I(q^k) = frac{sigma(q^k)}{q^k} = frac{q^{k+1} – 1}{q^k (q – 1)} < frac{q^{k+1}}{q^k (q – 1)} = frac{q}{q – 1}$$
from which it follows that
$$frac{2(q-1)}{q} < I(n^2) = frac{2}{I(q^k)} leq frac{2q}{q+1}.$$

Note that we then have the lower bound
$$I(n^2) > frac{2(q-1)}{q} geq frac{8}{5}$$
since $q$ is a prime satisfying $q equiv 1 pmod 4$.

Here is my initial question:

Can we improve the lower bound for $I(n^2)$ to
$$I(n^2) geq frac{5}{3}$$
using the following argument?

$$bigg(frac{2q}{q+1} geq I(n^2) > frac{5}{3}bigg) implies q > 5 implies q geq 13 implies bigg(I(n^2) > frac{2(q-1)}{q} geq frac{24}{13} > frac{5}{3}bigg)$$

Thus, we have the biconditional
$$I(n^2) > frac{5}{3} iff q > 5.$$

Next, we have the implication
$$I(n^2) = frac{5}{3} implies q = 5.$$

It then suffices to prove the implication
$$q = 5 implies I(n^2) = frac{5}{3}$$
to finally show that
$$I(n^2) geq frac{5}{3},$$
since $q geq 5$ holds.

But note that, if $q=5$, then
$$frac{5}{3} = I(n^2) = frac{2}{I(5^k)} = frac{2cdot{5^k}(5-1)}{5^{k+1}-1}$$
which implies that the Descartes-Frenicle-Sorli Conjecture that $k=1$ holds.

Still, notice that we have
$$k=1 implies I(q^k) = I(q) = frac{q+1}{q} = 1 + frac{1}{q} leq frac{6}{5} implies I(n^2) = frac{2}{I(q^k)} = frac{2}{I(q)} geq frac{2cdot{5}}{6} = frac{5}{3},$$
which is what we set out to prove.

Here is my final question:

Would it be possible to remove the reliance of the proof on the Descartes-Frenicle-Sorli Conjecture?

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